On meromorphic solutions of the equations related to the first Painlevé equation

Abstract

In this paper, we consider the generalised hierarchy of the first Painlevé equation which is a sequence of polynomial ordinary differential equations of even order that have a uniform differential-algebraic structure determined by the operator L~n. The first member of this hierarchy for n = 2 is the first Painlevé equation, and the subsequent equations of order 2n – 2 contain arbitrary parameters. They are named as higher analogues of the first Painlevé equation of 2n – 2 order. The article considers the analytical properties of solutions to the equations of the generalised hierarchy of the first Painlevé equation and the related linear equations. It is established that each hierarchy equation has one dominant term, and an arbitrary meromorphic solution of any hierarchy equation cannot have a finite number of poles. The character of the mobile poles of meromorphic solutions is determined. Using the Frobenius method, sufficient conditions are obtained for the meromorphicity of the general solution of the second-order linear equations with a linear potential defined by meromorphic solutions of the first three equations of the hierarchy.

Keywords

• the first painlevé equation
• hierarchies of painlevé equations
• meromorphic solutions